Renormalizations of branching random walks in equilibrium
The model studied in the paper is critical branching random walk on the \(d\)-dimensional lattice \(\mathbb{Z}^d\) with translation-invariant transition kernel, which in the case of transient symmetrized transition kernel is known to have non-degenerate equilibrium states. The author's goal is to investigate the large space structure of the equilibria, especially the fluctuation random field; more precisely, the objective is to find the right rescaling in order to obtain non-degenerate renormalization limits of large block averages of the particle configuration in an equilibrium state. The main result is that (under certain technical assumptions) in dimensions \(d\geq 3\), the fluctuation random field (centered around an ergodic translation-invariant limit) with rescaling \(r^{-(d+2)/2}\) converges as \(r\to\infty\) towards a Gaussian random field with kernel of the form const.\(/\overline Q(y- x)^{(d-2)/2}\), where \(\overline Q(x)\) is the quadratic form \(x^{tr}Q^{-1}x\) and \(Q\) is the matrix of second moments of the transition kernel of the random walk. The convergence is in the sense of weak convergence of \({\mathcal S}'(\mathbb{R}^d)\)-valued random elements, where \({\mathcal S}'(\mathbb{R}^d)\) is the nuclear space of tempered distributions (incorrectly stated to be a Banach space in the paper). An analogous space-time renormalization limit is also proved. The tools involve the historical process, which is used to work with a family decomposition, and Palm distributions.
scientific article; zbMATH DE number 1868586